PHYSICS 376 Final Examination 2014

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3. Ampere’s law.
One of Maxwell’s equations gives the curl of H~ as
r ⇥ H~ = J~ +
@
@t
D~ . (1)
a) Consider only the steady currents and fields so that the displacement current @D~
@t vanishes. Use Stokes’s theorem to show that this Maxwell equation (1) then gives Ampere’s law in the form
I
d~` · H~ = I , (2)
where the line integral H d~` · H~ is around a closed loop, and I is the current that passes through the surface enclosed by the loop.
b) Use the form (2) of Ampere’s law that you found in (a) to determine the magnetic field H~ at distance r from the axis of a long straight wire that carries a current I. (Here, r is greater than the radius of the wire, because only want the field in the region outside the wire itself.)
c) If the region outside the wire is empty space with permeability µ0, what is the magnetic field B~ (magnetic induction) at distance r from the wire?

4. A magnetized sphere.
A sphere of radius a is uniformly magnetized with a magnetization M~ = Mzˆ, where M is a constant.
a) Find the (volume) magnetic pole density ⇢M within the sphere.
b) Find the surface magnetic pole density (M everywhere on the surface of the sphere. ((M is also called the surface density of magnetic pole strength.)
c) Evaluate the total magnetic pole strength of this magnetized sphere. (This is the sum of the appropriate volume and surface integrals of ⇢M and (M.)
(You may wish to recall the analogies between ⇢M and (M and the similar quantities defined for electric polarization.)

 

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